# Abelian groups

Groups + xy = yx

## Examples

• Integers modulo n, (under addition modulo n), which is notated variously as C_n, Z_n, Z/(n). For example, C_4 has the multiplication table
```                 | 0 1 2 3
-+--------
0| 0 1 2 3
1| 1 2 3 0
2| 2 3 0 1
3| 3 0 1 2
```
• The free abelian group on k generators is isomorphic to Z^k, the direct product of k copies of the integers.

## Structure

Every finitely generated abelian group is isomorphic to some
```              Z_{q_1} x Z_{q_2} x ... x Z_{q_n} x Z^r
```
where q_1 divides q_2 divides ... divides q_n and r >= 0. r is called the rank of the abelian group, and q_i the invariants. The rank and invariants are uniquely determined by the group. For example, there are exactly six nonisomorphic abelian groups of order 360 = 2^3 . 3^2 . 5, namely
```             Z_360
Z_2 x Z_180
Z_2 x Z_2 x Z_90
Z_2 x Z_6 x Z_30
Z_3 x Z_120
Z_6 x Z_60
```

## Decision problems

Identity problem: Solvable
Word problem: Solvable

## Spectra and growth

Finite spectrum: 1,1,1,2,1,1,1,3,2,1,...
Free spectrum: 1, aleph_null, aleph_null, ...
Growth series: ((1+z)/(1-z))^r for the free abelian group on r generators

[Kurosh]

## Subsystems

A Catalogue of Algebraic Systems / John Pedersen / jfp@math.usf.edu